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- W4380714596 abstract "The similarity problem is one of the most famous open problems in the theory of $C^*$-algebras. We say that a $C^*$-algebra $cl A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $ucolon cl Ato cl B(H)$ is similar to a $*$-homomorphism and that a von Neumann algebra $cl A$ satisfies the weak similarity property ((WSP) for short) if every $mathrm{w}^*$-conitnuous unital and bounded homomorphism $ucolon cl Ato cl B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $cl A$ satisfies (WSP) if and only if the algebras $cl A^{prime}bar otimes cl B(ell^2(I))$ are hyperreflexive for all cardinals $I.$ In the case in which $cl A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $cl A^prime bar otimes cl B(ell^2(bb{N}))$ is hyperreflexive. We also introduce the hypothesis {bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {bf (CHH)}, all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $cl Abar otimes cl B,$ where $cl A$ is an injective von Neumann algebra and $cl B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the $text{w}^*$-similarity degree $d_{*}(cl Abar otimes cl B).$" @default.
- W4380714596 created "2023-06-15" @default.
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- W4380714596 date "2023-06-13" @default.
- W4380714596 modified "2023-10-17" @default.
- W4380714596 title "The similarity problem and hyperreflexivity of von Neumann algebras" @default.
- W4380714596 doi "https://doi.org/10.48550/arxiv.2306.07605" @default.
- W4380714596 hasPublicationYear "2023" @default.
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